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MULTISCALE SPARSE IMAGE REPRESENTATIONWITH LEARNED DICTIONARIES

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Niveau: Supérieur, Doctorat, Bac+8
MULTISCALE SPARSE IMAGE REPRESENTATIONWITH LEARNED DICTIONARIES Julien Mairal and Guillermo Sapiro Department of Electrical and Computer Engineering University of Minnesota, Minneapolis, MN 55455 Michael Elad Department of Computer Science Technion, Haifa 32000, Israel ABSTRACT This paper introduces a new framework for learning multiscale spa- rse representations of natural images with overcomplete dictionar- ies. Our work extends the K-SVD algorithm [1], which learns spa- rse single-scale dictionaries for natural images. Recent work has shown that the K-SVD can lead to state-of-the-art image restoration results [2, 3]. We show that these are further improved with a multi- scale approach, based on a Quadtree decomposition. Our framework provides an alternative to multiscale pre-defined dictionaries such as wavelets, curvelets, and contourlets, with dictionaries optimized for the data and application instead of pre-modelled ones. Index Terms— Image Restoration, Denoising, Multiscale, Sparsity 1. INTRODUCTION Consider a signal x ? Rn. We say that it admits a sparse approxima- tion over a dictionary D ? Rn?k, composed of k elements referred to as atoms, if one can find a linear combination of a “few” atoms from D that is “close” to the signal x. The so-called Sparseland model suggests that such dictionaries exist for various classes of sig- nals, and that the sparsity of a signal decomposition is a powerful model in many image processing applications [1, 2, 3].

  • patches

  • patch

  • improved using standard

  • patch-based image

  • patches simultaneously

  • multiscale sparse

  • results obtained


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MULTISCALE SPARSE IMAGE REPRESENTATION WITH LEARNED DICTIONARIES
Julien Mairal and Guillermo Sapiro Department of Electrical and Computer Engineering University of Minnesota, Minneapolis, MN 55455
ABSTRACT This paper introduces a new framework for learning multiscale spa rse representations of natural images with overcomplete dictionar ies. Ourwork extends the KSVD algorithm [1], which learns spa rse singlescale dictionaries for natural images.Recent work has shown that the KSVD can lead to stateoftheart image restoration results [2, 3]. We show that these are further improved with a multi scale approach, based on a Quadtree decomposition. Our framework provides an alternative to multiscale predefined dictionaries such as wavelets, curvelets, and contourlets, with dictionaries optimized for the data and application instead of premodelled ones. Index TermsImage Restoration, Denoising, Multiscale, Sparsity
1. INTRODUCTION
n Consider a signalxR. We say that it admits a sparse approxima n×k tion over a dictionaryDR, composed ofkelements referred to as atoms, if one can find a linear combination of a “few” atoms fromDthat is “close” to the signalx. ThesocalledSparseland model suggests that such dictionaries exist for various classes of sig nals, and that the sparsity of a signal decomposition is a powerful model in many image processing applications [1, 2, 3]. Another important assumption, commonly and successfully used in image processing, is the existence of multiscale features in images.Trying to design the best multiscale dictionary which fulfils a sparsity criterion has been a major challenge.Such at tempts include the wavelets, curvelets, contourlets, wedgelets, ban dlets, and steerable wavelets (see for example [4] and references therein). Thesemethods lead to many effective algorithms in im age processing, e.g., image denoising [5]. In [1] the KSVD is proposed for learning a singlescale dic tionary for sparse representation of image patches.By means of a sparsity prior on all fixedsized overlapping patches in the image, the KSVD is used for removing white Gaussian noise, leading to a highly efficient algorithm [2].This has been recently extended to color images, with stateoftheart results in denoising, inpainting, and demosaicing applications [3].In this paper, we extend the ba sic KSVD work, providing a framework for learning multiscale and sparse representation of images.In addition to the presentation of the new framework, we apply it to denoising, obtaining results that outperform reference works such as [2, 5, 6] and competes favorably with the most recent and stateoftheart in this field [7]. The task of learning a multiscale dictionary has been addressed in [8] in the general context of sparsifying image content.Our ap proach differs from this work in many ways, including:(i) their training algorithm employs a simple steepest descent while ours uses more effective iterations, thus leading to faster convergence; (ii) the
Work partially supported by NSF, ONR, NGA, DARPA, and the McK night Foundation.
Michael Elad Department of Computer Science Technion, Haifa 32000, Israel
structure of the multiscale process; and (iii) the way the found dic tionaries are deployed for denoising is entirely different, as we base our algorithm on the energy minimization method introduced in [2]. This explains the superior performance we obtain.
2. THESINGLESCALE KSVD DENOISING ALGORITHM
In this section, we briefly review the main ideas of the KSVD frame work for sparse image representation and denoising.The reader is referred to [1, 2, 3] for more details. Letx0be a clean image andy=x0+wits noisy version withwbeing an additive zeromean white Gaussian noise with a known standard deviationσ. The algorithm aims at finding a sparse √ √ approximation of everyn×noverlapping patch ofy, wherenis fixed apriori. This representation is done over an adapted dictionary D, learned for this set of patches. These approximations of patches are averaged to obtain the reconstruct image. This algorithm (shown in Figure 1) can be described as the minimization of an energy: ˘ ¯ 2 ˆ αˆij,D,ˆx= argminλ||xy||2(1) D,α ,x ij X X 2 +µij||αij||0+||DαijRijx||2. i,j ij
ˆ In this equation,xˆis the estimator ofx0, and the dictionaryDn×k Ris an estimator of the optimal dictionary which leads to the sparsest representation of the patches in the recovered image.The indices[i, j]mark the location of the patch in the image (represent k ing it’s topleft corner).The vectorsαˆijRare the sparse rep ˆ resentations for the[i, j]th patch inusing the dictionaryD. The 0 notation||.||0is thequasinorm, a sparsity measure, which counts the number of nonzero elements in a vector.The operatorRijis √ √ a binary matrix which extracts the squaren×npatch of coor dinates[i, j]from the image written as a column vector.The main steps of the algorithm are (refer to Figure 1): Sparse Coding Step:This is performed with an Orthogonal Match ing Pursuit (OMP) [9], which proves to be very efficient for diverse approximation problems [10].The approximation stops when the residual reaches a sphere of radiusnCσrepresenting the proba bility distribution of the noise. More on this is found in [3]. Dictionary Update:This is a sequence of onerank approximation problems that update both the dictionary atom and the sparse repre sentations that use it. Reconstruction:The last step is a simple averaging between the patches approximations and the noisy image.The denoised image isˆx. Equation (4) emerges directly from the energy minimization in Equation (2). Since it is well accepted that image information spreads across multiple scales, designing a KSVD type of algorithm that is able to adapt and capture information at multiple scales is the goal of this paper.